What is Cantilever Beam With Uniformly Varying Load?

Beam is a structural member whose lateral dimensions are much smaller than the longitudinal dimension and used to carry loads perpendicular to the longitudinal axis. The beams are generally used in buildings, bridges, trusses, etc., If a beam is subjected to a transverse load then the beam tends to get sheared off and bending stress is induced in the beam due to the bending moment.

Types of Beams

The beams may be classified based on the condition of end supports as follows:

Cantilever Beam

Simply Supported Beam

Overhanging Beam

Fixed Beam

Continuous Beam

Propped cantilever beam

Some Important Definitions

Shear Force

Bending Moment

Shear Force Diagram

Bending Moment Diagram

Shear Force

It is the algebraic summation of all the forces acting on the beam either left or right of the section of the beam. The sign convention for the shear force is considered as follows:

The forces acting on the left in the upward direction or right in the downward direction are considered as positive.

The forces acting on the left in the downward direction or right in the upward direction are considered as negative.

Bending Moment

It is the algebraic summation of the moments caused by all the forces acting on the beam either left or right of the section of the beam. The sign convention for the bending moment is considered as follows:

The sagging moments are considered as positive.

The hogging moments are considered as negative.

Shear Force Diagram

It is the diagram which represents the magnitude of the shear force at various points of the beam.

Bending Moment Diagram

It is the diagram which represents the magnitude of the bending moment caused by the forces acting at various points of the beam.

Cantilever Beam

If one end of the beam is fixed and the other end is free then the beam is known as Cantilever or Cantilever Beam as shown in figure. Cantilever beams are mostly used in bridges, trusses and other structures. When a concentrated load or point is applied on the cantilever beam at the free end, due to the distance “l” there will be mending moment as shown in figure and the beam hogs. This kind of moment is considered as negative.

Cantilever subjected to Uniformly Varying Load

Consider a cantilever beam of length L subjected to uniformly varying load or triangular load w N/m throughout its length as shown in figure.

Uniformly varying load is distributed over the entire span or part of the span in such a way that the intensity of the load gradually increases from zero to maximum.

Shear Force

Consider a section x-x at a distance of x from the free end.

Let, Fx and Mx be the shear force and bending moment at any section x-x respectively.

Now let us determine the rate of loading at the section x-x. The rate of loading is zero at the free end and gradually increases to w N/m at the fixed end. This means that the total load (w×x) is acting on the beam at the centre of gravity of loading. Therefore the shear force at the section x-x is given by (wx/L) per unit length. This quantity is shown in the load diagram.

The shear force at the section x-x is given by

Fx = Total load acting on the cantilever beam at the section x-x from the free end which will be equal to the area of the triangle BXC.

Fx = x (w x/L)(1/2) = (wx2/2L)

From the above equation it is very clear that the shear force increases from zero to (wx2/2L) according to parabolic law.

The shear force at the free end (that is, at x = 0)

FB = (wx2/2L) = 0

The shear force at the fixed end (that is, at x = L)

FB = (wx2/2L) = (wL2/2L) = (wL/2)

Bending Moment

The bending moment at the section x-x is given by

Mx = – (Total load acting on the beat for a length x) ×(Distance of the centre of gravity to the loading system to the section x-x)

= – (Area of the triangle) × (Distance of the centre of gravity to the loading system to the section x-x) – (wx2/2L) × (x/3) = – w x3/6L

The above equation shows that the bending moment increases gradually from zero to maximum according to the cubic law.

The bending moment at the free end (that is, at x = 0)

MB = 0

The bending moment at the fixed end (tha is, at x = L)

MA = Mmax = – w × L (l/3) = – w L 2/6

Example

A cantilever beam of length 6 metres carries an uniformly varying load which gradually increases from zero at the free end to a maximum of 3 kN/m at the fixed end. Draw the shear force and bending moment diagrams.

Given:

Length of the beam,

L = 6 m

Uniformly varying load,

w = 3 kN/m

Total load acting on the beam,

W = (1/2) × (w × L) = 9 kN

Shear Force

The shear force at the free end B is zero. The shear force at the fixed end is the area of the triangular load.

Shear force at the fixed end, FA = (1/2) × (w × L) = 9 kN

The shear force between the points B and A changes according to the parabolic law.

Bending Moment

The bending moment at the free end B is zero. The bending moment at the fixed end is equal to (– w L 2/6)

Therefore, MA = Mmax = = (– w L 2/6) = (– 3 × 6 2/6) = – 18 kN-m.

From the above equation it is evident that the bending moment increases gradually from zero to maximum according to the cubic law.